The Power Spectrum of the PSC Redshift Survey

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Mon. Not. R. Astron. Soc. 000, 000–000 (1999)

W. Sutherland1,2, H. Tadros3,1, G. Efstathiou4,C.S.Frenk5, O. Keeble6, S. Maddox4,R.G. 1 Dept. of Physics, Keble Road, Oxford OX1 3RH, UK 2 Visiting Observer, Cerro Tololo Interamerican Observatory 3 Astronomy Centre, University of Sussex, Brighton BN1 9QH, UK 4 Institute of Astronomy. Madingley Road, Cambridge CB3 0HA, UK 5 Dept. of Physics, South Road, Durham DH1 3LE, UK 6 Astrophysics Group, Imperial College, Blackett Laboratory, Prince Consort Road, London SW7 2BZ, UK 7 Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK 8 Max Planck Institute for Astrophysik, Karl-Schwarzschild-Strasse 1, D-8046 Garching bei Munchen, Germany.

Revised: WJS 16 Nov 98

ABSTRACT We measure the redshift-space power spectrum P (k) for the recently completed IRAS Point Source Catalogue (PSC) redshift survey, which contains 14500 galaxies over 84% of the sky with 60 micron flux 0.6 Jansky. Comparison with simulations shows that our estimated errors on P (k) are≥ realistic, and that systematic errors due to the finite > 1 survey volume are small for wavenumbers k 0.03 h Mpc− . At large scales our power spectrum is intermediate between those of the∼ earlier QDOT and 1.2 Jansky surveys, but with considerably smaller error bars; it falls slightly more steeply to smaller scales. We have fitted families of CDM-like models using the Peacock-Dodds formula for non-linear evolution; the results are somewhat sensitive to the assumed small-scale 1 velocity dispersion σV . Assuming a realistic σV 300 km s− yields a shape parameter ≈ 1 Γ 0.25 and normalisation bσ8 0.75; if σV is as high as 600 km s− then Γ = 0.5is only∼ marginally excluded. There∼ is little evidence for any ‘preferred scale’ in the power spectrum or non-Gaussian behaviour in the distribution of large-scale power. Key words: surveys – large-scale structure of Universe – galaxies: distances and redshifts

1INTRODUCTION present-day power spectrum must show a turnover to this < 1 slope at k 0.02 h Mpc− , close to the largest scales acces- As is well known (e.g. Peebles 1980), the power spectrum of sible to current∼ galaxy surveys. There is marginal evidence the galaxy distribution on large scales is of great importance for such a turnover in the APM data (Baugh & Efstathiou for testing cosmological models, since it can be related to the 1994a; Maddox, Efstathiou & Sutherland 1996; Tadros, Ef- initial conditions by linear perturbation theory. The power stathiou & Dalton 1998). spectrum has been estimated from a variety of galaxy redshift surveys, notably the CfA redshift survey (Park et al. Also, it is valuable to measure the power spectrum from 1994), the QDOT survey (Feldman, Kaiser & Peacock 1993, surveys with different selection criteria (e.g. optical & IRAS hereafter FKP), the Las Campanas redshift survey (Lin et selection). This is of considerable interest since the observed al. 1996), and the 1.2 Jy IRAS survey (Fisher et al. 1993). power spectrum is measured from the density field of galax- Also, the real-space power spectrum has been inferred from ies, whereas theory predicts the power spectrum of the mass the APM Galaxy Survey (Baugh & Efstathiou 1993; Baugh distribution. The process of galaxy formation is poorly un- & Efstathiou 1994a) by inversion of both the angular corre- derstood, so the observed Pg (k) may differ from Pm(k), pos- lation function and 2-D power spectrum. sibly in a complex way; indeed, since it appears that IRAS Despite these substantial surveys, there are still consid- galaxies and optical galaxies have different small-scale cor- erable uncertainties in the shape of the power spectrum on relation amplitudes, at least one of these cannot trace the large scales, since most of these surveys contain only a small mass. A simple ‘linear bias’ model is often assumed, in which number of independent structures, while the largest one (Las δg = bδm for some constant ‘bias factor’ b which may depend Campanas) has a slice-like geometry which complicates the on galaxy type; this model predicts that P (k) for optical estimation of the power spectrum. If the primordial power and IRAS galaxies should differ by a multiplicative factor n 2 spectrum is P (k) k with n 1 as suggested by infla- of (bO /bI ) . Such a model is reasonable since it has been tion, then for consistency∝ with the≈ COBE DMR results the shown by several authors (e.g. Fry & Gaztanaga 1993; Cole 2 W. Sutherland et al. et al. 1998; Mann et al. 1998) that if the galaxy density is a and UKST sky survey plates. If the APM or COSMOS data (possibly complex and stochastic) function only of the local showed no ‘obvious’ galaxy candidate near the IRAS source, < 1 mass density on scales 1 h− Mpc, then the effective bias we visually inspected the plate and attempted to classify the ∼ parameter defined by b(r) ξg(r)/ξm(r) tends to a con- source, rejecting it if it showed an obvious Galactic counter- stant on large scales, so such≡ a ‘local’ bias cannot alter the part e.g. an HII region, planetary nebula, dark cloud etc. We large-scale shape of the powerp spectrum. also exclude very faint galaxies (BJ > 19.5) from the redshift Another motivation for measuring the power spectrum survey since measuring their redshifts is time-consuming, from a large sample of IRAS galaxies is that there appears to and they are usually at z>0.1 and hence have little effect be a marginal discrepancy between the power spectra from on most of the desired analyses. the previous QDOT (Feldman, Kaiser & Peacock 1993, here- The sky coverage of the survey is the whole sky, exclud- after FKP) and 1.2 Jansky (Fisher et al. 1993) IRAS surveys, ing areas with less than 2 HCONs in the IRAS data, regions with the amplitude of P (k) from QDOT being roughly a fac- with optical extinction AV > 1.42 mag as estimated from tor of 2 higher at large scales. A counts-in-cells comparison the IRAS 100µm maps, and two small areas near the LMC of the two surveys does not reveal any obvious systematic er- and SMC. The resulting coverage is 84% of the whole sky. rors (Efstathiou 1995), but it is interesting to check whether (An extension to 93% sky coverage is in progress, using a these differences are consistent with sampling fluctuations in combination of K-band snapshots and HI redshifts). one or both surveys. Our 2-D source catalogue contains 17060 IRAS sources In this paper, we estimate the redshift-space power in the unmasked sky. Of these, 1593 are rejected as objects spectrum from a new redshift survey (Saunders et al. 1996) in our own Galaxy (e.g. cirrus, bright stars, reflection nebu- of some 14,500 galaxies over 84% of the sky, selected at 60µm lae, planetary nebulae etc), or as multiple entries from very from the IRAS Point Source Catalog (PSC). A variety of nearby galaxies ‘broken up’ by the IRAS point source de- analyses from this survey will appear shortly; the topology tection scheme. Another 648 sources are rejected either as of the density field has been analysed by Canavezes et al. very faint galaxies ( 400) or as sources without an optical ∼ (1998), the correlation function is analysed by Maddox et al. identification. This leaves 14819 galaxies in the ‘target’ list, (in preparation), the dipole is estimated by Rowan-Robinson and redshifts are now known for 14539 of these (98%). et al. (in preparation), a reconstruction of the peculiar veloc- Of these redshifts, 6500 are from a combination of ∼ ity field is given by Branchini et al. (1998), and the redshift- the 1.2 Jy survey (Fisher et al. 1995) and QDOT (Lawrence space distortions are estimated using spherical harmonics by et al. 1998), and 3000 are from other publications and ∼ Tadros et al. (in preparation). private communications. A further 4115 redshifts were mea- The plan of this paper is as follows: in 2wesum- sured by us for this survey, using 49 nights at the Isaac New- marise the construction and properties of the survey;§ in 3 ton Telescope, 18 nights at the Cerro Tololo 1.5-meter, and we present the power spectrum estimates, and we compare§ 6 nights at the Anglo-Australian Telescope, between 1992 these with results of N-body simulations in 4. In 5wecom- January and 1995 July. Details of the observations and data pare our results with other surveys and some§ parametrised§ reduction will be given elsewhere (Saunders et al.,inprepa- 1 cosmological models, and also set limits on non-Gaussian ration). The error on our redshifts is typically 150 km s− ; behaviour and periodicities. for the literature redshifts it is somewhat smaller. The me- 1 dian redshift of the sample is 8500 km s− , though there ≈ 1 is a long ‘tail’ extending to > 30000 km s− due to the broad luminosity function of IRAS galaxies. 2 THE PSC REDSHIFT SURVEY The construction of the PSC redshift survey (hereafter PSCz) is described in detail elsewhere (Saunders et al. 1996; 3 POWER SPECTRUM ESTIMATION Saunders et al. in preparation), but we summarise the main points here. The aim of the survey is to obtain redshifts for For the estimates here, we restrict the analysis to the un- o all galaxies with 60µm flux f60 > 0.6Jyoverasmuchofthe masked sky with the additional constraint b > 10 , since sky as feasible. The starting point for the survey is the QMW the survey may be slightly incomplete below| | this latitude; IRAS Galaxy Catalogue (Rowan-Robinson et al. 1991), but this gives a coverage of 78% of the full sky. We also set an 1 with modifications to extend the sky coverage and improve upper redshift limit of cz < 45000 km s− ,sincethesurveyis completeness. We have relaxed the IRAS colour criteria for incomplete at high redshift as noted above. This gives 13346 galaxy selection, and we have added in additional sources galaxies in the ‘default’ sample used for the power spectrum in the ‘2-HCON’ sky as follows: the IRAS satellite covered estimate. most of the sky with 3 hours-confirmed scans (HCONs) (Be- Since the geometry of the survey is well approximated ichman et al. 1988), while about 20% of the sky had only by a sphere, apart from the missing slice near the galac- 2 HCONs. Since a source must be detected in 2 separate tic plane, we follow the analysis of Feldman, Kaiser & Pea- HCONS for inclusion, the PSC catalogue may be less com- cock (1994, hereafter FKP) with minor modifications. This plete in the 2-HCON regions. Thus, in the 2-HCON sky we method provides an optimal weighting scheme with redshift added sources to our target list which had a 1-HCON detec- for estimating the power spectrum of an all-sky survey. More tion in the ‘Point Source Reject’ file and also had a matching sophisticated methods have been suggested by e.g. Tegmark entry in the IRAS Faint Source Catalog. (1995); these are very useful for surveys with highly non- These relaxed selection criteria allowed more contam- spherical geometries but are more complex than necessary ination of the target list by non-galaxy sources, but these for our survey. were excluded using APM or COSMOS scans of the POSS We convert the galaxy positions to comoving coordi- Power Spectrum of PSCz 3

3 3 Figure 1. Estimated redshift-space power spectra P (k)forweightsPe = 2000, 4000, 8000, 16000 h− Mpc . Open circles with error bars 3 3 show the result for each weight, while the solid line is the result for Pe = 8000 h− Mpc (the same in all panels). b nates assuming Ω0 = 1 and redshifts in the Local Group points respectively (Tadros & Efstathiou 1996), instead of 1 frame, and bin the galaxies in a cube of size 950 h− Mpc with α = Ng/Nr as in FKP (where Ng ,Nr are the numbers of 1283 cells. The FKP method assigns a redshift-dependent galaxies and randoms respectively). Secondly, we compute weight to each object, the shot noise using 1 ( )= (1) 2 2 2 w r Pshot = wi + α0 wj , (3) √A(1 + Pen(r)) g s X X where n(r) is the mean galaxy density at distance r, Pe is the estimated power spectrum (at some scale to be deter- where the two terms are the contributions from galaxies and mined), and A is a normalisation constant (see later). We random points respectively. The shot noise definition in FKP 2 use a parametric fit for the selection function determined us- Eq. 2.4.5 was Pshot = α(1 + α) s wj ; there the first-order ing the method of Springel & White (1998); here this takes term in α is the ‘expected’ shot noise from the galaxies given the form many realisations of the given selectionP function, while the first term in our definition is the ‘actual’ shot noise in the (1 α) γ (β/γ) n(z)=ny− /(1 + y ) (2) ∗ data. This makes negligible difference at large scales, but we y z/z ,z=0.0318 find from simulations that Eq. 3 gives substantially smaller ≡ ∗ ∗ α =1.769,β=4.531,γ=1.335 errors in the estimated power spectrum at small scales (large 3 3 3 k), because the shot noise term is substantial here and the n =8.76 10− h Mpc− ; ∗ × ‘actual’ shot noise from the galaxies may differ significantly these values are appropriate for f 0.6 Jy; for other flux from its expectation value estimated from the selection func- ≥ limits we simply scale z by (0.6/flim). tion. ∗ We have made three refinements to the FKP estimator: The third refinement is that we use a normalisation con- firstly, we define the ratio ofp densities of real and random vention given by Eq. A3; see Appendix A for a discussion of catalogues α0 = g wi/ s wj ,wherewi is the weight of the normalisation. the ith object and the sums run over galaxies and random The estimated power spectrum P (k) is then given as in P P b 4 W. Sutherland et al.

changing the ﬂux limit: the subsample with f60 > 0.8Jyhas a slightly lower amplitude of P (k) at all scales; the diﬀerence is mostly within the individual 1σ error bars but is evident over a range of k. The sliceb with 0.6

Z 2 3 3 2 P (k)= F(k) Pshot (6) P (k) =(2π)− dk0P(k0)G(k k0) (7) | |− h i | − | Z 1/2 where ng,ns are the number densities of galaxies and ran- − b 3 2 2 domb points in cubical cells, and P (k) is just the unweighted G(r) n(r)w(r) d r n (r)w (r) , ≡ average of P (k) over a spherical shell with mean radius k. Z The optimal weighting schemeb depends on the ac- and the normalisation is defined so that tual value of P (k), so the procedure is slightly cir- b 3 2 3 cular in principle. We have used values of Pe = d k G(k) =(2π) (8) 3 3 | | 2000, 4000, 8000, 16000 h− Mpc ; estimates of the power Z spectrum for each value of Pe are shown in Figure 1. by Parseval’s theorem. This convolution is a significant prob- We see that changing Pe changes the size of the error lem for slice-like survey geometries with a highly anisotropic bars, but there is little systematic difference in the resulting window function; but since our survey is large in all 3 di- estimates of P (k). This is as expected since FKP showed mensions, the window function is narrow. For our standard 3 3 that any choice of Pe gives an unbiased estimate of P (k) weighting with Pe = 8000 h− Mpc , the window function 2 (apart from theb convolution effects at small k discussed be- G(k) is illustrated in Figure 3 for three axes in Galac- low), but just weights different redshift shells differently - tic| coordinates,| and the angle-average. The window func- 2 2 larger Pe gives relatively more weight to more distant shells. tion is roughly approximated by a Gaussian exp( k /2k0 ) 3 3 1 − 4 We adopt Pe = 8000 h− Mpc as the default value for the with k0 0.006 h Mpc− at small k, with a roughly k− tail remainder of the paper. The resulting weights are illustrated arising from∼ the survey mask. Therefore, the effect of the in Figure 2: the solid line shows the weight function w(z), convolution on our estimates is small except at the largest < 1 and the dotted line shows the real-space window function scales k 0.03 h Mpc− . This is illustrated in Figure 4; the n(z)w(z). Also shown are the differential and cumulative solid lines∼ show the fractional contribution to the measured contributions to the survey ‘effective volume’ per unit red- P (k) as a function of ‘true’ wavenumber k0, i.e. Eq. 7 av- shift. eraged over directions of k, k0 for five values of observed 1 We have explored varying many of the selection criteria, kb=0.033, 0.066, 0.1, 0.2, 0.3 h Mpc− , assuming a CDM-like e.g. varying the model of the selection function, using a max- model P (k0) with the parameter Γ = 0.3(seeEq.11be- 1 imum redshift of 30000 km s− , or a galactic latitude cut of low). We have not attempted a deconvolution here, since 20o, and changing the flux threshold. Most of these changes the convolution effect is only important at small k where have a negligible effect on the results, with the exception of the estimates are becoming noisy. Power Spectrum of PSCz 5

Figure 3. The k-space window function G(k) 2. Points show this quantity for k parallel to the Galactic x, y, z directions (as labelled). | | 2 2 The solid line shows the direction-average over spherical shells of radius k. The dashed line shows a Gaussian exp( k /2k0)with 1 − k0=0.007 h Mpc− (this is for illustration and is not a ﬁt).

Figure 4. The eﬀect of convolution, i.e. the contribution to ‘observed’ power P (k) from ‘true’ wavenumber k0 inEq.7,summedover 1 directions of k, k0, per unit ln k0. Values are shown for observed k =0.033, 0.066, 0.1, 0.2, 0.3 h Mpc− (dashed lines). The y-scale is arbitrary. b 6 W. Sutherland et al.

Another eﬀect which causes the measured P (k)todevi- ate systematically from its true value is that the mean density of galaxies is not known independently butb is estimated from the survey. This leads to the constraint P (0) = 0, and the convolution above means that P (k) will also be under- estimated for small but non-zero k.Thiseﬀectwasnotedbyb Peacock & Nicholson (1991), and hasb been evaluated ana- lytically by Tadros & Efstathiou (1996) for the special case of a volume-limited survey; their Eq. A4.2 gives n P (k) P (k) W(k)2 (9) h − i≈−V| |

2 2 2 b n c W (k0) P (k0) W (k) − " | | # | | Xk0 For unequal weights as here, thec expression is complexc and best evaluated numerically; we have computed this for two models of P (k); a Γ = 0.3 CDM model, and an n = 1.2 1 − power-law, with a bend to n =0atk 0.01 h Mpc− .For each model, we generate a Gaussian random≤ density field with the assumed P (k), multiply by the PSCz window function n(r)w(r), set the mean of the windowed density field Figure 5. The effect of convolution and the normalisation con- to zero, and compute the power spectrum of the resulting dition P (k) = 0 on two model power spectra (solid lines); the upper solid line shows a power law P (k) k 1.2 with a break density field using the given w(r). The mean of the recov- ∝ − at k =0.01 h Mpc 1; the lower solid line shows Γ = 0.3CDM. ered power spectra from 30 realisations of each model is − The dashedb and dot-dash lines show the mean recovered power shown in Figure 5. We see that the bias is serious only for spectra for the two cases. Points show the measured values for < 1 k 0.02 h Mpc− ; note in particular that for the power-law PSCz. model,∼ the recovered power spectrum lies well above the 1 PSCz measurements at k 0.01 0.03 h Mpc− . The data ∼ − 1 points at k 0.013and0.02 h Mpc− are noteworthy here; 4 COMPARISON WITH SIMULATIONS although their∼ error bars appear large due to the log scale, none of the 30 realisations of the power-law model gave P (k) As a check of the code, and to assess whether the resulting as low as the PSCz data at these k. error bars are realistic, we have generated simulated ‘PSCz’ Thus, we can be confident that the flattening of the surveys from large N-body simulations from 3 cosmological 1 b observed power spectrum below k 0.06 h Mpc− is a real models and computed their power spectra as above. The 3 ∼ feature, not an artefact of the finite volume or normalisation. models are ‘standard’ CDM (SCDM), CDM with a cosmo- We compute the covariance matrix of P (k)using logical constant (ΛCDM), and a mixed cold + hot dark mat- Eq 2.5.2 of FKP. In practice, it is infeasible to evaluate this ter model (MDM); the simulations use a P3M code (Croft directly since it contains N 6 terms where N = 128; thus in & Efstathiou 1994) and their parameters are listed in Ta- ∼ b practice, we assign each of the N 3 wavevectors to its bin in ble 2. We place an ‘observer’ at a random location in the 6 k;foreachk, k0 we pick 10 random pairs of wavevectors cube, wrap the simulation using periodic boundary condi- ∼ k, k0 in the appropriate bins, and evaluate the sum accord- tions, and then select ‘galaxies’ in redshift space as random ingly. It is found that 106 pairs are necessary to ensure particles using the selection function of Eq. 2 and a ‘mask’ ∼ that the random errors are small, otherwise the resulting of the same shape as the real PSCz mask. We generated matrix can become non-positive-definite. a total of 27 simulated ‘PSCz’ surveys for each model; for A final effect on P (k) is the ‘binning factor’ noted by each model we used 9 different runs of the P3M code, and 3 Baugh & Efstathiou (1994b), which causes an underestimate different observer locations for each run. of small-scale power dueb to the galaxies being binned into Figure 6 shows the power spectra of the full box for finite-size cells before the Fourier transform. This ‘smooths’ each simulation, compared with results from 9 simulated the observed density field over the bin size. The size of this PSCz surveys. As expected, our estimator P (k)recov- effect depends on the slope of the true power spectrum; a ers the ‘true’ redshift-space power spectrum quite well for > 1 second-order approximation is given by Peacock & Dodds k 0.02 h Mpc− ; the simulated surveys slightly underes- 1 b (1996) Eq. 20, and is P (k)/P (k) 1+(kl)2/12 − ,where timate∼ the true redshift-space P (k) on intermediate scales ≈ lis the size of the unit cells in the FFT; this becomes inac- due to the convolution with the survey window function, but curate for kl > 2. We find that for n 1abetterapprox- the effect is small. Figure 7 shows the mean of the FKP er- b ∼− imation useful∼ for kl < π is ror bars from 9 simulations (triangles), compared with the 1 ‘real’ uncertainty estimated from the rms scatter between P (k)/P (k) 1+(kl)2/12 (kl)4/220 − . (10) ≈ − the 9 simulations (crosses). Clearly the FKP error estimates Since this correction is slightly model-dependent, we are a reasonable approximation to the ‘real’ errors in P (k), haveb not applied it to the estimates in Figure 1 or Figure 6, though they appear to underestimate the actual errors by but we have applied it before the model fitting in the fol- 20%. b lowing Section. ∼ In Figure 8 we compare the data to the mean of 27 sim- Power Spectrum of PSCz 7

Figure 7. Error estimates for simulated PSCz surveys, with the observed mask and selection function, for the three models in Table 2. Triangles show the mean of the FKP error estimate δP (k) from 9 simulated surveys; crosses show the ‘true’ error in P (k) estimated from the scatter in the 9 simulated surveys. b b

Figure 8. The observed PSCz power spectrum (points with 1σ errors) compared to the mean power spectrum of 27 simulated PSCz surveys for each of the 3 models (lines). 8 W. Sutherland et al.

Table 1. Estimated power spectrum

k P (k)Error k P(k) Error 1 3 3 1 3 3 ( h Mpc− )(h−Mpc )(hMpc− )(h−Mpc ) 0.0066 16110b 21200 0.0839 8690b 1165 0.0132 3981 5774 0.1056 4548 625 0.0198 3503 4039 0.1329 3786 425 0.0265 13514 5960 0.1674 2591 285 0.0334 14189 4861 0.2107 1466 193 0.0420 12459 3428 0.2653 1122 151 0.0529 10241 2321 0.3339 748 119 0.0666 8059 1488 0.4204 545 89

3 3 This table shows the estimated power spectrum P (k) with weight Pe = 8000 h− Mpc , and associated 1σ errors. These data 1 points have been ‘corrected’ for ﬁnite-size bins using Eq. 10 with l = 950/128 h− Mpc. Covariances between consecutive > 1 entries are substantial for small k but negligible forb k 0.1 h Mpc− . ∼

Table 2. N-body Simulation Parameters

1 Name N Box ( h− Mpc) ΩCDM ΩHDM ΩΛ hσ8 SCDM 1603 600 1 0 0 0.5 1.0 ΛCDM 1603 600 0.2 0 0.8 1.0 1.0 MDM 1003 300 0.7 0.3 0 0.5 0.67

ulated PSCz surveys for each model. We find that all three scales our measurements are well within the range of previ- 1 models give a reasonable match to the shape of the observed ous surveys, but on intermediate scales 0.2 h Mpc− our power spectrum; the ΛCDM model has somewhat too high measurements are slightly steeper than the∼ others, notably an amplitude and would require an antibias b<1. However, the combined QDOT and 1.2 Jy surveys. This is the cause this could be remedied by lowering h somewhat and raising of the low values of Γ 0.2 for best-fitting CDM-like mod- ∼ Ω0, keeping Γ constant; this would reduce the implied σ8 for els seen in the next section. It appears that optical galaxies COBE normalisation, and improve the fit. Alternatively, a have a somewhat higher power spectrum amplitude on inter- modest degree of ‘tilt’ with primordial spectral index n<1 mediate scales, as expected from the fact that IR-selected would similarly reduce the COBE-normalised σ8. surveys contain mainly late-type galaxies, and from their Although the SCDM model has much less large-scale smaller correlation length r0, but it remains unclear whether power than the others in real space, the high COBE normal- this persists to large scales; a direct comparison with APM is isation gives two effects: an enhancement of power on large complicated by the redshift-space distortion, and the inter- scales by a factor 1.86 from the Kaiser (1987) redshift- pretation of LCRS is somewhat complicated by the inversion space distortion, and≈ a suppression of small-scale power from the 2-D to 3-D power spectrum due to the slice-like ge- from the resulting large peculiar velocities. These two ef- ometry. Future large optical surveys such as 2dF and Sloan fects combine to bring the redshift-space P (k)ofthismodel should greatly clarify this question. into rather good agreement with the data. These effects for COBE-normalised SCDM have been previously noted by various authors, e.g. Bahcall et al. (1993), although of 5.2 Fits to the power spectrum course this model has serious problems with cluster abun- In addition to the direct comparison with simulations, it dances (Eke, Cole & Frenk 1996), large-separation gravita- is interesting to extract best-fitting values for parametrised tional lenses (Cen et al. 1994) etc. models of the power spectrum; we use firstly linear theory for simplicity, and later the fitting formulae of Peacock & Dodds (1996) which account both for non-linear evolution 5 DISCUSSION of clustering, and the effect of distortions between real space and redshift space. 5.1 Comparison with Other Surveys We use CDM-like models with the initial power spec- Our observed redshift-space power spectrum for Pe = trum parametrised by Γ as in Eq. 7 of Efstathiou, Bond & 3 3 8000 h− Mpc is compared with a number of previous mea- White (1993, hereafter EBW), which is surements in Figure 9. These come from various catalogues, Bk both optical and IRAS selected; all are redshift-space power P (k)= (11) 3/2 2 ν 2/ν spectra except for the APM data which come from an in- (1 + [ak +(bk) +(ck) ] ) 1 1 version of the 2-D power spectrum. We see that on large a =6.4/Γh−Mpc,b=3.0/Γh− Mpc, Power Spectrum of PSCz 9

Figure 9. Comparison of PSCz P (k) with other measured power spectra. The data points as labelled on the ﬁgure are: IRAS 1.2 Jansky from Fisher et al. (1993), QDOT from FKP, QDOT + 1.2 Jansky from Tadros & Efstathiou (1995), Stromlo-APM from Tadros & Efstathiou (1996), APM (real-space, deconvolved) from Baugh & Efstathiou (1994), and Las Campanas (deconvolved) from Lin et al.(1996). For clarity, error bars areb only shown on PSCz and APM (others are larger), and Stromlo, QDOT and QDOT+1.2 Jansky data have been rebinned. Dotted lines show linear-theory CDM with Γ = 0.2and0.5withσ8=0.8.

1 c =1.7/Γh−Mpc ν =1.13 assume simple ‘linear bias’ so the galaxy power spectrum is b2 the matter power spectrum. There is clearly insuffi- In linear theory there are only 2 free parameters: the shape × cient information in the P (k) data to fit all 6 parameters parameter Γ, and the normalisation which may be taken as separately, so we restrict the parameter space as follows: bσ8,wherebis the bias parameter and σ8 is the rms mass 1 2 fluctuation in an 8 h− Mpc top-hat sphere. The χ contours (a) To set Ω, we consider either Einstein-de Sitter mod- using the full covariance matrix are shown in Figure 11; the els, with Ω = 1, ΩΛ = 0, treating Γ as a free param- best fit values are Γ = 0.19 0.03 and bσ8 =0.80 0.02. eter (which is a reasonable approximation to e.g. mixed ± ± These compare to Γ = 0.19 0.06,bσ8 =0.87 0.07 for the darkmattermodels);orweconsiderΛCDMmodelswith ± ± QDOT survey, as given by FKP Eq. 4.3.3. Ω=Γ/0.66, ΩΛ =1 Ω (i.e. assuming Γ = Ωh with a − For the non-linear Peacock-Dodds formula, we need a Hubble constant h =0.66, consistent with most recent mea- total of 6 parameters to specify the present-day redshift surements); space power spectrum: the initial mass power spectrum is (b) To set σ8, we use either the cluster normalisation specified as above by Γ and σ8 (where σ8 is defined as ( 0.52+0.13Ω) σ =0.52Ω − (Eke, Cole & Frenk 1996), or we use the initial rms mass fluctuation, multiplied by the linear- 8 COBE normalisation where σ is a function of Γ using Eq. 6 theory growth factor to the present day). The subsequent 8 of EBW and Qrms =17µK. non-linear evolution depends also on Ω0 and (weakly) on 1 ΩΛ. The transformation from real to redshift space depends (c) To set σV ,wefixeitherσV = 300 or 600 km s− ,or on Ω0 and the bias b, and also on the pairwise peculiar ve- predict σV as a function of Γ,σ8 using the fitting formula in locity dispersion; in the PD formula this is approximated by Eqs. 40a, 40b of Mo, Jing & Borner (1997, hereafter MJB). 1 assuming galaxy velocities are independent Gaussians with A value as high as 600 km s− is probably disfavoured by 2 dispersion σV = v12(r)/2 for some suitable scale r.We observations (Landy, Szalay & Broadhurst 1998), but we p 10 W. Sutherland et al.

Table 3. Fits of CDM-like models

1 ΩNorm(σ8)σV(kms− )Γ bσ8 Plot? Linear — — 0.19 0.80 * 1 COBE (0.45) 300 0.20 0.67 1 COBE (0.49) MJB (368) 0.22 0.68 1 COBE (1.00) 600 0.40 0.71 1 Clus (0.52) 300 0.20 0.66 1 Clus (0.52) MJB (390) 0.23 0.69 1 Clus (0.52) 600 0.34 0.81 Γ/0.66 COBE (0.97) 300 0.16 0.66 Γ/0.66 COBE (0.97) MJB (457) 0.16 0.70 Γ/0.66 COBE (1.38) 600 0.34 0.73 Γ/0.66 Clus (0.91) 300 0.20 0.67 * Γ/0.66 Clus (0.81) MJB (438) 0.25 0.74 * Γ/0.66 Clus (0.73) 600 0.31 0.83 *

Fits to the observed P (k)inthe(Γ,bσ8) plane. The first line shows linear theory, the remainder use the non-linear PD formula, using various choices for setting the other input parameters (Ω, ΩΛ,σ8,σV ) as a function of Γ. Column 1: either Ω = 1, ΩΛ =0 or Ω = 0.66/Γ, ΩΛ =1 Ω; Column 2: σ8 defined either by COBE or cluster normalisation; Column 3: σV either fixed to 300 1 b − or 600 km s− , or defined as a function of the other parameters using the MJB model. For the σ8 and σV columns, the values in brackets show the derived values at the best-fitting Γ. The * in the last column flags the models plotted in Figs. 10 and 11. include this as a conservative upper limit to show the effect For ΛCDM models, the value of Γ at which the COBE on the derived parameters. and cluster normalisations agree is Γ = 0.17,σ8 1.0; this ≈ Having made one choice from each of (a),(b),(c) above, is interestingly close to our best-fit values of Γ 0.2. Our fit ∼ this defines Ω Ω as a function of Γ; we then treat values of bσ8 thus imply b 0.75: this is a significant amount , Λ,σ8,σV ∼ Γandbas free parameters, and fit to the observed P (k) of “antibias”, but may be plausibly accounted for by the data. We choose to use only the data points in the range deficiency of IRAS galaxies in rich clusters. Thus a ΛCDM 1 model is attractive in that it can simultaneously satisfy three 0.021

The best-fitting values of Γ,bσ8 for each of the above parameter choices are shown in Table 3. (We present the fit 5.3 Periodicities and Spikes results in terms of (Γ,bσ8) rather than (Γ,b) because b and σ8 are degenerate in the linear regime, and also for simplicity There have been a number of suggestions of a ‘preferred 2 since the contours of equal χ are roughly parallel to the axes scale’ for large-scale clustering, notably by Broadhurst et al. in the (Γ,bσ8) plane.) For the low-Ω cluster-normalised case, (1990), Landy et al. (1996), and Einasto et al. (1997). These contours of goodness of fit are shown in Figure 11 for each effects, if real, could arise from a ‘spike’ in the power spec- of the three σV assumptions. The derived power spectra for trum, such as may arise from a baryon isocurvature model the best fits in the same cases are shown in Figure 10. For or non-standard inflation models; or from non-Gaussian ini- most of the fits, a fairly small value of Γ 0.2isfavoured, tial conditions, which could lead to one particular direction ∼ and Γ 0.5 is quite strongly ruled out; for the cases with showing a value of δ(k) 2 much larger than its expectation ∼ 1 | | σV = 600 km s− , higher values Γ 0.35 are favoured and value P (k); or even from an intrinsic ‘preferred direction’ in ∼ Γ=0.5 is only marginally ruled out, though values of σV as the Universe. high as this are definitely not favoured observationally. In our data, there is marginal evidence for a ‘spike’, per- There is rather little difference between the goodness- haps better described as a ‘step’, in the power spectrum near 1 of-fit for the various choices, though the fits for σV = k 0.08 h Mpc− , but this is only about a 2σ effect above a 1 ∼ 600 km s− are somewhat worse. For a given parameter smooth CDM-like fit, and the scale is significantly different 1 choice, the 1σ random errors are typically 15% in Γ and from that k 0.06 h Mpc− suggested by Broadhurst et al. ± ∼ 4% in bσ8. In general we see that the systematic uncer- (1990) and Landy et al. (1996); also, we find that changing ± tainties from the different choices of Ω,σ8,σV dominate the our flux limit to f60 > 0.7 Jy causes a substantial drop in random errors arising from the error bars on P ; the largest P (k) at this point, while leaving other points virtually un- source of uncertainty is that arising from σV .Asexpected, changed. Thus, we suspect this point may be a statistical if we increase σV the predicted small-scale powerb decreases fluctuation,b and there is no conclusive evidence for a feature at a fixed Γ, and thus the best-fitting values of Γ and bσ8 in our power spectrum. From inspection of results from a increase to compensate. number of realisations of N-body simulations, we find that Power Spectrum of PSCz 11

2 Figure 11. Contours of χ in the (Γ,bσ8) plane. of ﬁts to the observed power spectrum. The upper left panel shows linear theory. Other panels use the Peacock-Dodds formula as in Sec. 5.2, with Ω = Γ/0.66, cluster normalisation, and diﬀerent choices of random velocities 2 2 2 σV as labelled. The cross denotes the minimum χ , contours are at χ = χmin +1,4,9,16, 25.

2 2 3 3 such features quite commonly arise from statistical fluctua- variance σ = a /2=(2π)− d kP(k) gives the constant tions. C =(2π)3a2/4. Then, after convolution with the survey win- To test for non-Gaussian effects, we have examined the dow function as in Eq. 7, thisR contributes to the observed histogram of the ratio of observed to mean power power spectrum a term of size 2 F (k) /(P (k)+Pshot) for each wavenumber; for Gaussian | | 2 clustering, this should follow an exponential distribution a 2 2 Pspike(k)= G(k k0) + G(k+k0) . (12) with unit mean. Results for several ranges of wavenumber 4 | − | | | are shown in Figure 12, and closely follow the exponential Recall that distribution. The presence of the shot noise somewhat weak- b > 3 2 ens this test, since for smaller scales Pshot P (k), but at 2 d rn(r)w(r) ∼ G(k =0) = ; (13) large scales this is a sensitive test for non-Gaussian initial | | d3rn2(r)w2(r) conditions with a ‘tail’ to high values. The distributions are R well fitted by the exponential, so there is no evidence for for a volume-limitedR survey with constant n, w this just non-Gaussian initial conditions or any ‘preferred direction’ reduces to the survey volume V , so it may be thought in our survey. of as the survey ‘effective volume’. For our survey with 3 3 This also strongly constrains any ‘preferred direction’, weight function given by Pe = 8000 h− Mpc ,wehave 2 7 3 3 as follows. If there exists a strict plane-wave periodicity in G(0) =6.2 10 h− Mpc : thus even a small amplitude | | × the universe with dimensionless density contrast a along periodic wave of a =0.15 would lead to a large spike in our 5 3 3 wavevector k0, the true power spectrum contains delta func- measured P (k0) 3 10 h− Mpc ,welloutsidetheexpo- ≈ × tions at k0, P (k)=C[δ(k k0)+δ(k+k0)]. Requiring the nential tail of measured values. We conclude that there is no ± − b 12 W. Sutherland et al.

Figure 12. Solid lines show histograms of ‘observed’ power P (k)+Pshot divided by the mean for each k. The dashed line shows an exponential distribution with unit mean. The four panels show various ranges in k as labelled. b strictly periodic plane-wave structure in our survey volume spectrum, and no evidence for large-scale periodicity or non- with amplitude larger than 15%. Gaussianity.

Acknowledgements 6CONCLUSIONS We are very grateful to many observers, especially Marc Our conclusions may be summarised as follows: Davis, Tony Fairall, Karl Fisher & John Huchra for supply- (i) The redshift-space power spectrum of the PSCz sur- ing redshifts in advance of publication. We thank the staff at vey is intermediate between those of the earlier QDOT and the INT, AAT and CTIO telescopes for support, especially 1.2 Jansky IRAS surveys on large scales, though it is slightly Hernan Tirado and Patricio Ugarte at CTIO. W. Sutherland steeper on small scales. It is stable against variations in the is supported by a PPARC Advanced Fellowship. W. Saun- galactic cuts and redshift limits etc, though the amplitude ders is supported by a Royal Society Fellowship. GE thanks decreases slightly for a flux cut > 0 8Jy. f . PPARC for the award of a Senior Fellowship. (ii) There is convincing evidence∼ for curvature in the power spectrum; the slope changes from the small-scale < 1 power law n 1.4ton 0 on scales k 0.07 h Mpc− . This is not an≈− artefact of∼ the finite sample∼ volume or the APPENDIX A1: NORMALISATION OF P(K) estimation of the mean density from the survey. (iii) The best-fitting CDM-like models have Γ We note an issue concerning the overall normalisation of ∼ 0.25,bσ8 0.7. The uncertainties in these values are mainly P (k). FKP set a normalisation of their weight function via due to uncertainties∼ in the small-scale velocity dispersion, Eq. 2.4.1. It is convenient in practice to set weights via Eq. 1 the value of Ω etc, rather than statistical errors. with A = 1, and then later divide all power spectra by a (iv) There is little evidence for a ‘spike’ in the power constant A. There are several ways of doing this, which give Power Spectrum of PSCz 13

Figure 10. Fits to the measured power spectrum (points) using the Peacock-Dodds formula as in Sec. 5.2. Vertical dotted lines denote the range of k used in the fits. Lines show the fitted power spectra as labelled, one for linear theory and three for different values of σV , for the case of a low-density universe with Ω = Γ/0.66, and cluster normalisation.

2 2 2 α ik.ri 2 G(k) = cie wi ;(A4) | | A3V − cells s ! X X b requiring G(k ) 2 = 1 leads to Eq. A3.) k | | Figure 6. Power spectra for the 3 models in Table 2. For each The relationship between A1,A2,A3 is as follows: the model, the dotted and solid lines show the real and redshift space P number of randomb points Nr is arbitrarily fixed, and we power spectrum of the full simulation box (mean of 9 runs, error 3 define β = d r n(r)/Nr to be the ratio of the expected bars negligible). The circles show the mean of the estimated power number of galaxies from the selection function to the number spectra from 9 simulated ‘PSCz’ surveys, with 1σ error on the of randoms,R which should be similar but not identical to α. mean. Thus the expected number of randoms in cell i is just niv/β where ni n(ri). For small cells, all randoms in a given cell ≡ will have equal weight wi,thusci =wi Poisson(niv/β). similar but not identical results: The LHS of FKP Eq. 2.4.1 Thus × gives 2 A2 = α ci niwi 3 2 2 h i h i A1 = d r n (r)w (r); (A1) cells X 2 2 Z = α ni vwi /β the RHS gives cells X 2 α A = α n(rs)w (rs), (A2) = A1 (A5) 2 β s X 2 Since ci is a Poisson variable multiplied by wi, c = where the sum runs over random points, and = g r as i α N /N 2 2 h i before. Another possible definition is wi [(niv/β) + nv/β], thus Eq. A3 gives

2 2 2 α 2 2 α v 2 2 A3 = c w , (A3) A3 = wi ni v i − i h i v β2 cells s ! cells X X X where the ﬁrst sum runs over the cells used in the Fourier v 2 2 + wi ni (niv/β)wi transform, ci is the sum of weights of all random points in β − cells cells ! the ith cell and v is the volume of a unit cell. (This results X X α2v from estimating the window function from the FFT of the = w2n2 β2 i i random points via cells X 14 W. Sutherland et al.

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